Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Sextuple Angle
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Example of Use of Sine of Integer Multiple of Argument: Formulation 2
- $\map \sin {6 \theta } = \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta}$
Proof
\(\ds \map \sin {6 \theta}\) | \(=\) | \(\ds \cos^n \theta \paren {\dbinom n 1 \tan \theta - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}\) | Sine of Integer Multiple of Argument: Formulation 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos^6 \theta \paren {\dbinom 6 1 \tan \theta - \dbinom 6 3 \paren {\tan \theta}^3 + \dbinom 6 5 \paren {\tan \theta}^5 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta }\) |
$\blacksquare$